A note on the number of records near the maximum

Let {Xn,n[greater-or-equal, slanted]1} be a sequence of independent identically distributed random variables with the continuous distribution function F(x). Let Kn(a) denote the number of values j[set membership, variant]{1,2,...,n} for which Xj[set membership, variant](Mn-a,Mn], where Mn=max{X1,...,Xn} and a is a positive constant. In this paper we prove that limn-->[infinity] E(Kn(a))=1 if and only if Kn(a) converges in probability to one, if and only if when F(x) has a thick tail. Furthermore, we will give a necessary and sufficient condition for